Load Packages and Import Data

# To install a package, run the following ONCE (and only once on your computer)
# install.packages("psych")  
library(here)  # makes reading data more consistent
library(tidyverse)  # for data manipulation and plotting
library(haven)  # for importing SPSS/SAS/Stata data
library(lme4)  # for multilevel analysis
library(sjPlot)  # for plotting effects
library(MuMIn)  # for computing r-squared
># Warning: package 'MuMIn' was built under R version 4.0.5
library(broom.mixed)  # for summarizing results
library(modelsummary)  # for making tables
library(interactions)  # for plotting interactions
theme_set(theme_bw())  # Theme; just my personal preference
msummary_mixed <- function(models, coef_map = NULL, ...) {
  if (is.null(coef_map)) {
    if (!"list" %in% class(models)) {
      models <- list(models)
    }
    for (model in models) {
      coef_map <- union(coef_map, tidy(model)$term)
    }
    ranef_index <- grep("^(sd|cor)__", x = coef_map)
    coef_map <- c(coef_map[-ranef_index], coef_map[ranef_index])
    names(coef_map) <- coef_map
  } else {
    ranef_index <- grep("^(sd|cor)__", x = names(coef_map))
  }
  rows <- data.frame(term = c("Fixed Effects", "Random Effects"))
  rows <- cbind(rows, rbind(
    rep("", length(models)),
    rep("", length(models))
  ))
  length_fes <- length(coef_map) - length(ranef_index)
  if ("statistic" %in% names(list(...)) && is.null(list(...)$statistic)) {
    attr(rows, "position") <- c(1, (length_fes + 1))
  } else {
    attr(rows, "position") <- c(1, (length_fes + 1) * 2)
  }
  modelsummary::msummary(models, coef_map = coef_map, add_rows = rows, ...)
}

Import Data

# Read in the data (pay attention to the directory)
hsball <- read_sav("hsball.sav")

Between-Within Decomposition

Cluster-mean centering

To separate the effects of a lv-1 predictor into different levels, one needs to first center the predictor on the cluster means:

hsball <- hsball %>% 
  group_by(id) %>%   # operate within schools
  mutate(ses_cm = mean(ses),   # create cluster means 'ses_cm' (the same as `meanses`)
         ses_cmc = ses - ses_cm) %>%   # cluster-mean centered 'ses_cmc'
  ungroup()  # exit the "editing within groups" mode
# The meanses variable already exists in the original data, but it's slightly
# different from computing it by hand
hsball %>% 
  select(id, ses, meanses, ses_cm, ses_cmc)

Model Equation

Lv-1:

\[\text{mathach}_{ij} = \beta_{0j} + \beta_{1j} \text{ses_cmc}_{ij} + e_{ij}\]

Lv-2:

\[ \begin{aligned} \beta_{0j} & = \gamma_{00} + \gamma_{01} \text{meanses}_j + u_{0j} \\ \beta_{1j} & = \gamma_{10} \end{aligned} \] where

  • \(\gamma_{10}\) = regression coefficient of student-level ses representing the expected difference in student achievement between two students in the same school with one unit difference in ses,
  • \(\gamma_{01}\) = between-school effect, which is the expected difference in mean achievement between two schools with one unit difference in meanses.

Running in R

We can specify the model as:

m_bw <- lmer(mathach ~ meanses + ses_cmc + (1 | id), data = hsball)

You can summarize the results using

summary(m_bw)
># Linear mixed model fit by REML ['lmerMod']
># Formula: mathach ~ meanses + ses_cmc + (1 | id)
>#    Data: hsball
># 
># REML criterion at convergence: 46568.6
># 
># Scaled residuals: 
>#     Min      1Q  Median      3Q     Max 
># -3.1666 -0.7253  0.0174  0.7557  2.9453 
># 
># Random effects:
>#  Groups   Name        Variance Std.Dev.
>#  id       (Intercept)  2.692   1.641   
>#  Residual             37.019   6.084   
># Number of obs: 7185, groups:  id, 160
># 
># Fixed effects:
>#             Estimate Std. Error t value
># (Intercept)  12.6481     0.1494   84.68
># meanses       5.8662     0.3617   16.22
># ses_cmc       2.1912     0.1087   20.16
># 
># Correlation of Fixed Effects:
>#         (Intr) meanss
># meanses -0.004       
># ses_cmc  0.000  0.000

Contextual Effect

With a level-1 predictor like ses, which has both student-level and school-level variance, we can include both the level-1 variable and the school mean variable as predictors. When the level-1 predictor is present, the coefficient for the group mean variable becomes the contextual effect.

Model Equation

Lv-1:

\[\text{mathach}_{ij} = \beta_{0j} + \beta_{1j} \text{ses}_{ij} + e_{ij}\]

Lv-2:

\[ \begin{aligned} \beta_{0j} & = \gamma_{00} + \gamma_{01} \text{meanses}_j + u_{0j} \\ \beta_{1j} & = \gamma_{10} \end{aligned} \] where

  • \(\gamma_{10}\) = regression coefficient of student-level ses representing the expected difference in student achievement between two students in the same school with one unit difference in ses,
  • \(\gamma_{01}\) = contextual effect, which is the expected difference in student achievement between two students with the same ses but from two schools with one unit difference in meanses.

Running in R

We can specify the model as:

contextual <- lmer(mathach ~ meanses + ses + (1 | id), data = hsball)

You can summarize the results using

summary(contextual)
># Linear mixed model fit by REML ['lmerMod']
># Formula: mathach ~ meanses + ses + (1 | id)
>#    Data: hsball
># 
># REML criterion at convergence: 46568.6
># 
># Scaled residuals: 
>#     Min      1Q  Median      3Q     Max 
># -3.1666 -0.7253  0.0174  0.7557  2.9453 
># 
># Random effects:
>#  Groups   Name        Variance Std.Dev.
>#  id       (Intercept)  2.692   1.641   
>#  Residual             37.019   6.084   
># Number of obs: 7185, groups:  id, 160
># 
># Fixed effects:
>#             Estimate Std. Error t value
># (Intercept)  12.6613     0.1494  84.763
># meanses       3.6750     0.3777   9.731
># ses           2.1912     0.1087  20.164
># 
># Correlation of Fixed Effects:
>#         (Intr) meanss
># meanses -0.005       
># ses      0.004 -0.288

If you compare the REML criterion at convergence number you can see this is the same as the between-within model. The estimated contextual effect is the coefficient of meanses minus the coefficient of ses_cmc, which is the same as what you will get in the contextual effect model.

Random-Coefficients Model

Explore Variations in Slopes

hsball %>%
  # randomly sample 16 schools
  filter(id %in% sample(unique(id), 16)) %>%
  ggplot(aes(x = ses, y = mathach)) +
  geom_point(size = 0.5) +
  geom_smooth(method = "lm") +
  facet_wrap(~id)
># `geom_smooth()` using formula 'y ~ x'

Model Equation

Lv-1:

\[\text{mathach}_{ij} = \beta_{0j} + \beta_{1j} \text{ses_cmc}_{ij} + e_{ij}\]

Lv-2:

\[ \begin{aligned} \beta_{0j} & = \gamma_{00} + \gamma_{01} \text{meanses}_j + u_{0j} \\ \beta_{1j} & = \gamma_{10} + u_{1j} \end{aligned} \] The additional term is \(u_{1j}\), which represents the deviation of the slope of school \(j\) from the average slope (i.e., \(\gamma_{10}\)).

Running in R

We have to put ses in the random part:

ran_slp <- lmer(mathach ~ meanses + ses_cmc + (ses_cmc | id), data = hsball)

You can summarize the results using

summary(ran_slp)
># Linear mixed model fit by REML ['lmerMod']
># Formula: mathach ~ meanses + ses_cmc + (ses_cmc | id)
>#    Data: hsball
># 
># REML criterion at convergence: 46557.7
># 
># Scaled residuals: 
>#     Min      1Q  Median      3Q     Max 
># -3.1768 -0.7269  0.0160  0.7560  2.9406 
># 
># Random effects:
>#  Groups   Name        Variance Std.Dev. Corr 
>#  id       (Intercept)  2.6931  1.6411        
>#           ses_cmc      0.6858  0.8281   -0.19
>#  Residual             36.7132  6.0591        
># Number of obs: 7185, groups:  id, 160
># 
># Fixed effects:
>#             Estimate Std. Error t value
># (Intercept)  12.6454     0.1492   84.74
># meanses       5.8963     0.3600   16.38
># ses_cmc       2.1913     0.1280   17.11
># 
># Correlation of Fixed Effects:
>#         (Intr) meanss
># meanses -0.004       
># ses_cmc -0.088  0.004

Plotting Random Slopes

augment(ran_slp, data = hsball) %>% # augmented data (adding EB estimates)
  ggplot(aes(x = ses, y = .fitted, color = factor(id))) +
  geom_smooth(method = "lm", se = FALSE, size = 0.5) +
  labs(y = "Predicted mathach") +
  guides(color = "none")
># `geom_smooth()` using formula 'y ~ x'

Including the effect of meanses

# augmented data (adding EB estimates)
augment(ran_slp, data = hsball) %>%
  ggplot(aes(x = ses, y = mathach, color = factor(id))) +
  # Add points
  geom_point(size = 0.2, alpha = 0.2) +
  # Add within-cluster lines
  geom_smooth(aes(y = .fitted),
              method = "lm", se = FALSE, size = 0.5) +
  # Add group means
  stat_summary(aes(x = meanses, y = .fitted,
                   fill = factor(id)),
               color = "red",  # add border
               fun = mean, 
               geom = "point",
               shape = 24,
               # use triangles
               size = 2.5) +
  # Add between coefficient
  geom_smooth(aes(x = meanses, y = .fitted),
              method = "lm", se = FALSE,
              color = "black") +
  labs(y = "Math Achievement") +
  # Suppress legend
  guides(color = "none", fill = "none")
># `geom_smooth()` using formula 'y ~ x'
># `geom_smooth()` using formula 'y ~ x'

Or on separate graphs:

# Create a common base graph
pbase <- augment(ran_slp, data = hsball) %>%
  ggplot(aes(x = ses, y = mathach, color = factor(id))) +
  # Add points
  geom_point(size = 0.2, alpha = 0.2) +
  labs(y = "Math Achievement") +
  # Suppress legend
  guides(color = "none")
# Lv-1 effect
p1 <- pbase + 
  # Add within-cluster lines
  geom_smooth(aes(y = .fitted),
              method = "lm", se = FALSE, size = 0.5)
# Lv-2 effect
p2 <- pbase +
  # Add group means
  stat_summary(aes(x = meanses, y = .fitted),
               fun = mean,
               geom = "point",
               shape = 17,
               # use triangles
               size = 2.5) +
  # Add between coefficient
  geom_smooth(aes(x = meanses, y = .fitted),
              method = "lm", se = FALSE,
              color = "black")
# Put the two graphs together (need the gridExtra package)
gridExtra::grid.arrange(p1, p2, ncol = 2)  # in two columns
># `geom_smooth()` using formula 'y ~ x'
># `geom_smooth()` using formula 'y ~ x'

Effect Size

As discussed in the previous week, we can obtain the effect size (\(R^2\)) for the model:

(r2_ran_slp <- r.squaredGLMM(ran_slp))
># Warning: 'r.squaredGLMM' now calculates a revised statistic. See the help page.
>#            R2m       R2c
># [1,] 0.1684161 0.2310862

Analyzing Cross-Level Interactions

sector is added to the level-2 intercept and slope equations

Model Equation

Lv-1:

\[\text{mathach}_{ij} = \beta_{0j} + \beta_{1j} \text{ses_cmc}_{ij} + e_{ij}\]

Lv-2:

\[ \begin{aligned} \beta_{0j} & = \gamma_{00} + \gamma_{01} \text{meanses}_j + \gamma_{02} \text{sector}_j + u_{0j} \\ \beta_{1j} & = \gamma_{10} + \gamma_{11} \text{sector}_j + u_{1j} \end{aligned} \] where - \(\gamma_{02}\) = regression coefficient of school-level sector variable representing the expected difference in achievement between two students with the same SES level and from two schools with the same school-level SES, but one is a catholic school and the other a private school. - \(\gamma_{11}\) = cross-level interaction coefficient of the expected slope difference between a catholic and a private school with the same school-level SES.

Running in R

We have to put the sector * ses interaction to the fixed part:

# The first step is optional, but let's recode sector into a factor variable
hsball <- hsball %>% 
  mutate(sector = factor(sector, levels = c(0, 1), 
                         labels = c("Public", "Catholic")))
crlv_int <- lmer(mathach ~ meanses + sector * ses_cmc + (ses_cmc | id), 
                 data = hsball)

You can summarize the results using

summary(crlv_int)
># Linear mixed model fit by REML ['lmerMod']
># Formula: mathach ~ meanses + sector * ses_cmc + (ses_cmc | id)
>#    Data: hsball
># 
># REML criterion at convergence: 46514.5
># 
># Scaled residuals: 
>#      Min       1Q   Median       3Q      Max 
># -3.11560 -0.72905  0.01592  0.75280  3.01543 
># 
># Random effects:
>#  Groups   Name        Variance Std.Dev. Corr
>#  id       (Intercept)  2.3807  1.5430       
>#           ses_cmc      0.2716  0.5212   0.24
>#  Residual             36.7077  6.0587       
># Number of obs: 7185, groups:  id, 160
># 
># Fixed effects:
>#                        Estimate Std. Error t value
># (Intercept)             12.0846     0.1987   60.81
># meanses                  5.2450     0.3682   14.24
># sectorCatholic           1.2523     0.3062    4.09
># ses_cmc                  2.7877     0.1559   17.89
># sectorCatholic:ses_cmc  -1.3478     0.2348   -5.74
># 
># Correlation of Fixed Effects:
>#             (Intr) meanss sctrCt ss_cmc
># meanses      0.245                     
># sectorCthlc -0.697 -0.355              
># ses_cmc      0.071 -0.002 -0.046       
># sctrCthlc:_ -0.048 -0.001  0.071 -0.664

Effect Size

By comparing the \(R^2\) of this model (with interaction) and the previous model (without the interaction), we can obtain the proportion of variance accounted for by the cross-level interaction:

(r2_crlv_int <- r.squaredGLMM(crlv_int))
>#            R2m       R2c
># [1,] 0.1759122 0.2284431
# Change
r2_crlv_int - r2_ran_slp
>#              R2m          R2c
># [1,] 0.007496143 -0.002643047

Therefore, the cross-level interaction between sector and SES accounted for an effect size of \(\Delta R^2\) = 0.7%. This is usually considered a small effect size (i.e., < 1%), but interpreting the magnitude of an effect size needs to take into account the area of research, the importance of the outcome, and the effect sizes when using other predictors to predict the same outcome.

Simple Slopes

With a cross-level interaction, the slope between ses_cmc and mathach depends on a moderator, sector. To find out what the model suggests to be the slope at a particular level of sector, which is also called a simple slope in the literature, you just need to look at the equation for \(\beta_{1j}\), which shows that the predicted slope is \[\hat \beta_{1} = \hat \gamma_{10} + \hat \gamma_{11} \text{sector}.\] So when sector = 0 (public school), the simple slope is \[\hat \beta_{1} = \hat \gamma_{10} + \hat \gamma_{11} (0),\] or 2.7877417. When sector = 1 (private school), the simple slope is \[\hat \beta_{1} = \hat \gamma_{10} + \hat \gamma_{11} (1),\] or 2.7877417 + (-1.3477638)(1) = 1.4399779.

Plotting the Interactions

crlv_int %>%
  augment(data = hsball) %>%
  ggplot(aes(
    x = ses, y = .fitted, group = factor(id),
    color = factor(sector)  # use `sector` for coloring lines
  )) +
  geom_smooth(method = "lm", se = FALSE, size = 0.5) +
  labs(y = "Predicted mathach", color = "sector")
># `geom_smooth()` using formula 'y ~ x'

You can also use the interactions::interact_plot() function for just the fixed effects:

interact_plot(crlv_int, 
              pred = "ses_cmc",
              modx = "sector",
              modx.labels = c("Public", "Catholic"), 
              plot.points = TRUE, 
              point.size = 0.5, 
              point.alpha = 0.2, 
              facet.modx = TRUE,  # use this to make two panels
              x.label = "SES (cluster-mean centered)", 
              y.label = "Math achievement")

Summaizing Analyses in a Table

You can again use the msummary() function from the modelsummary package to quickly summarize the results in a table. However, there is an issue as the order of the coefficients are messed up, as shown below.

# Basic table from `modelsummary` (ordering messed up)
msummary(list(
  "Between-within" = m_bw,
  "Contextual" = contextual,
  "Random slope" = ran_slp,
  "Cross-level interaction" = crlv_int
))
Between-within Contextual Random slope Cross-level interaction
(Intercept) 12.648 12.661 12.645 12.085
(0.149) (0.149) (0.149) (0.199)
meanses 5.866 3.675 5.896 5.245
(0.362) (0.378) (0.360) (0.368)
ses_cmc 2.191 2.191 2.788
(0.109) (0.128) (0.156)
sd__(Intercept) 1.641 1.641 1.641 1.543
sd__Observation 6.084 6.084 6.059 6.059
ses 2.191
(0.109)
cor__(Intercept).ses_cmc −0.195 0.244
sd__ses_cmc 0.828 0.521
sectorCatholic 1.252
(0.306)
sectorCatholic × ses_cmc −1.348
(0.235)
Num.Obs. 7185 7185 7185 7185
AIC 46578.6 46578.6 46571.7 46532.5
BIC 46613.0 46613.0 46619.8 46594.4
Log.Lik. −23284.289 −23284.290 −23278.825 −23257.264
REMLcrit 46568.577 46568.580 46557.651 46514.529

I’ve made a function msummary_mixed(), defined in the beginning of this page, as a hack to get a nicer table. Basically, you can just replace msummary() with msummary_mixed(), with everything else being the same:

# Basic table using `msummary_mixed`
msummary_mixed(list(
  "Between-within" = m_bw,
  "Contextual" = contextual,
  "Random slope" = ran_slp,
  "Cross-level interaction" = crlv_int
))
Between-within Contextual Random slope Cross-level interaction
Fixed Effects
(Intercept) 12.648 12.661 12.645 12.085
(0.149) (0.149) (0.149) (0.199)
meanses 5.866 3.675 5.896 5.245
(0.362) (0.378) (0.360) (0.368)
ses_cmc 2.191 2.191 2.788
(0.109) (0.128) (0.156)
ses 2.191
(0.109)
sectorCatholic 1.252
(0.306)
sectorCatholic:ses_cmc −1.348
(0.235)
Random Effects
sd__(Intercept) 1.641 1.641 1.641 1.543
sd__Observation 6.084 6.084 6.059 6.059
cor__(Intercept).ses_cmc −0.195 0.244
sd__ses_cmc 0.828 0.521
Num.Obs. 7185 7185 7185 7185
AIC 46578.6 46578.6 46571.7 46532.5
BIC 46613.0 46613.0 46619.8 46594.4
Log.Lik. −23284.289 −23284.290 −23278.825 −23257.264
REMLcrit 46568.577 46568.580 46557.651 46514.529

Bonus: More “APA-like” Table

Make it look like the table at https://apastyle.apa.org/style-grammar-guidelines/tables-figures/sample-tables#regression. You can see a recent recommendation in this paper published in the Journal of Memory and Language

# Step 1. Create a map for the predictors and the terms in the model.
# Need to use '\\( \\)' to show math.
# Rename and reorder the rows. Need to use '\\( \\)' to
# show math. If this does not work for you, don't worry about it.
cm <- c("(Intercept)" = "Intercept",
        "meanses" = "school mean SES",
        "ses_cmc" = "SES (cluster-mean centered)",
        "sectorCatholic" = "Catholic school",
        "sectorCatholic:ses_cmc" = "Catholic school x SES (cmc)",
        "sd__(Intercept)" = "\\(\\tau_0\\)",
        "sd__ses_cmc" = "\\(\\tau_1\\) (SES)",
        "sd__Observation" = "\\(\\sigma\\)")
# Step 2. Create a list of models to have one column for coef, 
# one for SE, one for CI, and one for p
models <- list(
  "Estimate" = crlv_int,
  "SE" = crlv_int,
  "95% CI" = crlv_int
)
# Step 3. Add rows to say fixed and random effects
# (Need same number of columns as the table)
new_rows <- data.frame(
  term = c("Fixed effects", "Random effects"),
  est = c("", ""),
  se = c("", ""),
  ci = c("", "")
)
# Specify which rows to add
attr(new_rows, "position") <- c(1, 7)
# Step 4. Render the table
msummary(models,
         estimate = c("estimate", "std.error",
                      "[{conf.low}, {conf.high}]"),
         statistic = NULL,  # suppress the extra rows for SEs
         coef_map = cm,
         add_rows = new_rows)
Estimate SE 95% CI
Fixed effects
Intercept 12.085 0.199 [11.695, 12.474]
school mean SES 5.245 0.368 [4.523, 5.967]
SES (cluster-mean centered) 2.788 0.156 [2.482, 3.093]
Catholic school 1.252 0.306 [0.652, 1.852]
Catholic school x SES (cmc) −1.348 0.235 [−1.808, −0.888]
Random effects
\(\tau_0\) 1.543
\(\tau_1\) (SES) 0.521
\(\sigma\) 6.059
Num.Obs. 7185 7185 7185
AIC 46532.5 46532.5 46532.5
BIC 46594.4 46594.4 46594.4
Log.Lik. −23257.264 −23257.264 −23257.264
REMLcrit 46514.529 46514.529 46514.529